Branch-width and path-width are width parameters of graphs and matroids, which measure how easy it is to decompose a graph or a matroid into a tree-like or path-like structure via separations of small order. These parameters have been used not only for designing efficient algorithms with the inputs of small branch-width or path-width, but also for proving theoretical structural theorems by providing a rough structural description. We will describe a polynomial-time algorithm to construct a path-decomposition or a branch-decomposition of width at most k, if it exists, for a matroid represented over a fixed finite field for fixed k. Our approach is based on the dynamic programming combined with the idea developed by Bodlaender for his work on tree-width of graphs. For path-width, this is a new result. For branch-width, this improves the previous work by Hlineny and Oum (Finding branch-decompositions and rank-decompositions, SIAM J. Comput., 2008) which was very indirect; their algorithm is based on the upper bound on the size of minor obstructions proved by Geelen et al. (Obstructions to branch-decompositions of matroids, JCTB, 2006) and requires testing minors for each of these obstructions. Our new algorithm does not use minor obstructions. As a corollary, for graphs, we obtain an algorithm to construct a rank-decomposition of width at most k if it exists for fixed k. This is a joint work with Jisu Jeong (KAIST) and Eun Jung Kim (CNRS-LAMSADE).

We prove that every infinite sequence of skew-symmetric or symmetric matrices M1, M2, … over a fixed finite field must have a pair Mi, Mj (i<j) such that that Mi is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in Mj, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour’s theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle’s theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum’s theorem for graphs of bounded rank-width with respect to pivot-minors.

A non-complete distance-regular graph Γ is called geometric if there exists a set C of Delsarte cliques such that each edge of Γ lies in a unique clique in C. In this talk, we determine the non-complete distance-regular graphs satisfying max{3,8(a1+1)/3}<k<4a1+10−6c2. To prove this result, we first show by considering non-existence of 4-claws that any non-complete distance-regular graph satisfying max{3,8(a1+1)/3}<k<4a1+10−6c2 is a geometric distance-regular graph with smallest eigenvalue −3. Moreover, we classify the geometric distance-regular graphs with smallest eigenvalue −3. As an application, 7 feasible intersection arrays are ruled out.

The H-colouring Dichotomy of Hell and Nesetril, proved in 1990, is one of the most quoted results in the field of Graph Homomorphisms. It says that H-coloring, the problem of deciding if a given graph G admits an homomorphism to the fixed graph H, is NP-complete if H contains an odd cycle, and otherwise polynomial time solvable.
In this talk we present a short new proof of this result, recently published, using a new projective property defined for homomorphisms of powers of a graph G onto a graph H.

A classical theorem of Cedric Smith guarantees the existence of a second Hamilton circuit other than a given one in any hamiltonian cubic graph. It is an open problem in complexity theory whether the corresponding search problem is polynomially solvable. We observe that a search algorithm, implicit in Bill Tutte’s nonconstructive proof of Smith’s theorem, has exponential running time. We also mention two possible candidates for search algorithms with polynomial complexity.

A sequence of even length is a repetition if the first half is identical to the second half. A sequence is said to contain a repetition if it has a subsequence which is a repetition. A classical result of Thue says that there is an infinite sequence on 3 symbols which contains no repetition. This result motivated many deep research and challenging problems. One graph concept related to this result is Thue-colouring. A Thue-colouring of a graph G is a mapping which assigns to each vertex of G a colour (a symbol) in such a way that the colour sequence of any path of G contains no repetition. The Thue-chromatic number of a graph is the minimum number of colours needed in a Thue-colouring of G. Thue’s result is equivalent to say that the infinite path has Thue-chromatic number 3. It is also known that the Thue-chromatic number of any tree is at most 4.
Thue-choice number of a graph G is the list version of its Thue-chromatic number, which is the minimum integer k such that if each vertex of G is given k-permissible colours, then there is a Thue-colouring of G using a permissible colour for each vertex. This talk will survey some research related to Thue Theorem and will show that Thue-choice number of paths is at most 4 and Thue choice number of trees are unbounded.

Lovász and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2cn perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2n/18 perfect matchings, thus verifying the conjecture for claw-free graphs.